Dynamics of a Simple Model for Turbulence of the Second Sound in Helium II

Abstract

The results of numerical experiments on chaotic ('turbulent') dynamics of the second sound in helium II are presented and discussed based on a very simple model proposed and theoretically studied recently by Khalatnikov and Kroyter. Using a powerful present-day techniques for the studying nonlinear phenomena, we confirm their results on the stationary oscillation in helium and its stability as well as on a qualitative picture of successive transitions to limit cycles and chaos. However, the experiments revealed also a much more complicated structure of the bifurcations than it was expected. The fractal structure of chaotic attractors was also studied including their noninteger dimension. Surprisingly, a very simple model used in all these studies not only qualitatively represents the behavior of helium in laboratory experiments but also allows for a correct order-of-magnitude estimate of the critical heat pumping into helium at bifurcations.